Currently Available:
• Character sums to smooth moduli are small, published in Canad. J. Math. 62 (5), 2010, 1099--1115.
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• Multiplicative mimicry and improvements of the Pólya-Vinogradov inequality, published in Algebra and Number Theory Vol. 6 (2012), No. 1, 123--163.
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• Lower bounds on odd order character sums (joint with Youness Lamzouri), published in IMRN 2012 (21), 5006--5013.
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• A quadratic large sieve inequality over number fields (joint with Benoît Louvel), published in Math. Proc. Camb. Phil. Soc., Volume 154 (02), 2013, 193--212.
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• The distribution of the maximum of character sums (joint with Jonathan Bober), published in Mathematika, Vol. 59 (02), 2013, pp. 427--442.
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• L-functions with n-th order twists (joint with Valentin Blomer and Benoît Louvel), published in IMRN 2014 (7), 1925--1955.
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• Large even order character sums (joint with Youness Lamzouri), published in Proc. Amer. Math. Soc. 142 (2014) no. 8, 2609--2614.
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• On the spectral distribution of large weighted random regular graphs (joint with C. Khoury, S. J. Miller, and K. Ninsuwan), Random Matrices Theory Appl. 3 (2014), no. 4, 1450015, 22 pp.
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• Pólya-Vinogradov and the least quadratic nonresidue (joint with J. W. Bober), to appear in Math. Ann.
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• The frequency and structure of large character sums (joint with J. W. Bober, A. J. Granville, and D. Koukoulopoulos), submitted.


• Postbreakthrough behavior in flow through porous media (Eduardo Lopez et al.), published in Physical Review E 67 (2003).
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Coming Soon(ish):

• An Erdős-Kac theorem for the number of prime factors of multiplicative orders (joint with G. Martin), in preparation.
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• A density result for character sums (joint with A. J. Harper, Y. Lamzouri), in preparation.

Summaries:

A classical theorem of Paley asserts the existence of an infinite family of quadratic characters whose character sums become exceptionally large. In this paper, we establish an analogous result for characters of any fixed even order. Previously our bounds were only known under the assumption of the Generalized Riemann Hypothesis.

A natural quantity associated to a Dirichlet character χ (mod q) is M(χ), the maximum modulus attained by character sums of χ. It is known that √q << M(χ) << √q log q. Moreover, Montgomery and Vaughan have shown that for most characters, M(χ) << √q. We refine their methods and investigate the distribution of M(χ).

In 1932, Paley constructed an infinite family of quadratic Dirichlet characters whose character sums get exceptionally large. This lower bound is optimal, in view of an upper bound of the same quality proved by Montgomery and Vaughan under the assumption of the Generalized Riemann Hypothesis. One can ask whether Paley's construction can be extended to higher-order characters. Working under the GRH, Granville and Soundararajan did so for all characters of even order. In this paper, we unconditionally prove an analogue of Paley's result for characters of odd order. Our result is optimal, in view of a conditional upper bound I proved earlier.

In this paper I study exponential sums whose coefficients are multiplicative and belong to the complex unit disc. The main result asserts that such a sum has substantial cancellation unless the coefficient function f(n) is essentially a Dirichlet character twisted by n^it, for some real t. As an application I refine recent work of A. Granville and Soundararajan on character sums. Among other consequences, conditionally on the Generalized Riemann Hypothesis I derive an upper bound for odd-order character sums which is best-possible.

I demonstrate how non-trivial bounds on very short character sums, combined with recent work of A. Granville and Soundararajan, can be used to improve the classical Pólya-Vinogradov inequality for long character sums. In particular, I obtain new upper bounds on character sums for characters to powerful or smooth moduli.

Using numerical and analytic methods, we determine the probability density of the travel time between two given points in both homogenous and inhomogenous media, the latter modeled via percolation theory. The results have potential applications to oil recovery.

In 1995, Heath-Brown introduced a quadratic large sieve, which has turned out to be a very useful tool. Onodera recently generalized his sieve to the quadratic extension Q(i). In this paper, we generalize Heath-Brown's result to all number fields. One of the principal difficulties is to find an appropriate formulation of the theorem in the number field setting; we do so in terms of n-th order Hecke families, which we introduce and study.

Friedberg, Hoffstein, and Liemann introduced a family of double Dirichlet series, which are built out of the n^th-order twists of a fixed Hecke L-series (a closely related series was also studied by Diaconu and Tian). Among other nice properties, a typical member Z(s,w) of this family satisfies a functional equation taking (s,w) to (1−s,1−w). This gives rise to a `convexity' bound for Z(¹⁄₂+iu, ¹⁄₂+it). Inspired by recent work of Blomer, we establish a subconvexity bound in the (u,t)-aspect.

A celebrated result of Erdős and Kac from 1940 asserts that ω(n), the number of prime factors of n, behaves like a normally distributed random variable with mean and variance log log n. In 1985, Erdős and Pomerance conjectured that an analogous result should hold for ω(L_n(a)), where a is a fixed integer and L_n(a) is the multiplicative order of a (mod n). Their conjecture remains open, although it has been shown to follow from a suitable generalization of the Riemann Hypothesis. In the present paper, we unconditionally prove that the conjecture holds `on average' over a. We also derive several related results.

In the early '80s, B. D. McKay investigated and described the limiting spectral distribution of large, random, regular graphs. In this paper, we study a generalization of his problem. Given a random regular graph, weigh the edges of the graph by random variables drawn from a probability distribution w, and compute the corresponding spectral distribution. To every weight distribution w there corresponds an associated limiting spectral distribution Tw. In this paper, we classify the eigendistributions of the operator T.

In this note, we exhibit a new approach to bounding the least quadratic nonresidue via long character sums. Among other results, assuming a conjectured improvement of the Pólya-Vinogradov inequality, we prove that the least quadratic nonresidue (mod p) is bounded by (log p)^1.4 for all p = 3 (mod 4).

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